Let a1-a2-a3-a9 be in harmonic progression- If a4-5 and a5-4- then the value of 1-a1 1-a2 1-a31-a4 1-a5 1-a61-a7 1-a8 1-a9 is

A_1,a_2,a_3…,a_9 are in H.P. a_n=1a+(n 1)d a_4=1a+3d=5 ⇒ a+3d=15 ⋯(1) Also, nbsp;a_5=1a+4d=4 ⇒ a+4d=14 ⋯(2) Solving (1) and (2), we get nbsp; a=d=120 ∴ ...

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Byju's Answer

Standard XII

Mathematics

Integration by Substitution

Let a1,a2,a3…...

Question

Let $a_{1}, a_{2}, a_{3} \dots, a_{9}$ be in harmonic progression. If $a_{4} = 5$ and $a_{5} = 4,$ then the value of $∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 / a_{1} & 1 / a_{2} & 1 / a_{3} 1 / a_{4} & 1 / a_{5} & 1 / a_{6} 1 / a_{7} & 1 / a_{8} & 1 / a_{9} \end{matrix} ∣ ∣ ∣ ∣ ∣$ is

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Solution

$a_{1}, a_{2}, a_{3} \dots, a_{9}$ are in H.P.
$a_{n} = \frac{1}{a + (n - 1) d}$
$a_{4} = \frac{1}{a + 3 d} = 5$
$\Rightarrow a + 3 d = \frac{1}{5} \dots (1)$
Also, $a_{5} = \frac{1}{a + 4 d} = 4$
$\Rightarrow a + 4 d = \frac{1}{4} \dots (2)$
Solving $(1)$ and $(2),$ we get
$a = d = \frac{1}{20}$
$∴ a_{n} = \frac{20}{n} \Rightarrow \frac{1}{a_{n}} = \frac{n}{20}$

Now, let $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 / a_{1} & 1 / a_{2} & 1 / a_{3} 1 / a_{4} & 1 / a_{5} & 1 / a_{6} 1 / a_{7} & 1 / a_{8} & 1 / a_{9} \end{matrix} ∣ ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{20} & \frac{2}{20} & \frac{3}{20} \frac{4}{20} & \frac{5}{20} & \frac{6}{20} \frac{7}{20} & \frac{8}{20} & \frac{9}{20} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$

$Δ = \frac{1}{(20)^{3}} ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 \end{matrix} ∣ ∣ ∣ ∣$

Applying $R_{3} \to R_{3} - R_{2},$ then $R_{2} \to R_{2} - R_{1},$ we get
$Δ = \frac{1}{(20)^{3}} ∣ ∣ ∣ ∣ \begin{matrix} 1 & 2 & 3 3 & 3 & 3 3 & 3 & 3 \end{matrix} ∣ ∣ ∣ ∣ = 0$

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Similar questions

Q. If

(1 + x)^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{10} x^{10}

, then value

(a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})^{2} + (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2}

Q. If

{(1 + x)}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . . a_{10} x^{10}

, then value of

{(a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})}^{2} + {(a_{1} - a_{3} + a_{5} - a_{7} + a_{9})}^{2}

Q. If

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then

\frac{a_{1}}{a_{2} + a_{3}} + \frac{a_{2}}{a_{3} + a_{4}} + \frac{a_{3}}{a_{4} + a_{5}} + \frac{a_{4}}{a_{5} + a_{1}} + \frac{a_{5}}{a_{1} + a_{2}} \geq \frac{5}{2}

Q. If

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Q. If

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Let a1,a2,a3…,a9 be in harmonic progression. If a4=5 and a5=4, then the value of ∣∣ ∣ ∣∣1/a11/a21/a31/a41/a51/a61/a71/a81/a9∣∣ ∣ ∣∣ is

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